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Lesson 9.2 Ellipses

Lesson 9.2 Ellipses. Ellipse Set of all points where the sum of the distances to two fixed points ( foci ) is constant. 8cm. 7cm. 5cm. 4cm. 9cm. 3cm. 10cm. 2cm. Focus. Focus. Other parts of an ellipse:. Major axis. Center. Minor axis. Vertices.

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Lesson 9.2 Ellipses

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  1. Lesson 9.2Ellipses

  2. Ellipse Set of all points where the sum of the distances to two fixed points (foci) is constant. 8cm 7cm 5cm 4cm 9cm 3cm 10cm 2cm Focus Focus

  3. Other parts of an ellipse: Major axis Center Minor axis Vertices Major & Minor axis can run in opposite direction:

  4. Deriving the Equation for an Ellipse Equation comes from distances: b a c a→ distance from center to vertex on major axis b → distance from center to vertex on minor axis c → distance from center to a focus

  5. Relation among a, b, & c: Equation of an ellipse derives from distances: or

  6. USING THE EQUATION • 1) Center is at h and k • 2) a is the larger denominator/ b is smaller (major/minor) • Use a, b, and/or c to find missing values using the equations • Use a, b, and c distances to find foci and vertices from center • If a is under x, the ellipse “goes” Left/Right • If a is under y, the ellipse “goes” Up/Down

  7. Example Name the foci, center, length of major and minor axes. Then sketch the ellipse.

  8. Eccentricity where 0 < e < 1 small e → circular e close to 1 → more elliptical

  9. Example Find the center, foci, vertices and eccentricity. 9x2 + 4y2 + 36x – 24y + 36 = 0

  10. Example Write the equation for the ellipse with foci (0,0), (0,8); and major axis length of 16.

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